Question

If AD \(\perp\) BC in an equilateral triangle ABC, then AD\(^2\) will be

• A. \( \frac{1}{2} \text{CD}^2 \)
• B. \( 4 \text{CD}^2 \)
• C. \( 2 \text{CD}^2 \)
• D. \( 3 \text{CD}^2 \)

Hint:

For a 30-60-90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\). In \(\triangle ADC\), \(\angle C = 60^\circ\), \(\angle D = 90^\circ\), and \(\angle CAD = 30^\circ\). The sides are CD (opposite 30°), AD (opposite 60°), and AC (opposite 90°). Thus, AD = CD \(\times \sqrt{3}\). Squaring both sides gives AD\(^2\) = 3 CD\(^2\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In an equilateral triangle, the altitude from a vertex to the opposite side bisects the opposite side. This creates two congruent 30-60-90 right-angled triangles. We can use the Pythagorean theorem to establish a relationship between the altitude (AD) and the segment of the base (CD).
Step 2: Key Formula or Approach:
1. In \(\triangle ABC\), since it is equilateral, AB = BC = AC.
2. Since AD \(\perp\) BC, AD is an altitude. In an equilateral triangle, the altitude is also the median, so D is the midpoint of BC. Thus, CD = \( \frac{1}{2} \) BC.
3. Apply the Pythagorean theorem to the right-angled triangle \(\triangle ADC\): \( AC^2 = AD^2 + CD^2 \).
Step 3: Detailed Explanation:
Let the side length of the equilateral triangle be \(s\).
So, AB = BC = AC = \(s\).
Since D is the midpoint of BC, we have CD = \( \frac{1}{2} \) BC = \( \frac{s}{2} \).
Now, consider the right-angled triangle \(\triangle ADC\).
According to the Pythagorean theorem:
\[ AC^2 = AD^2 + CD^2 \] Substitute the values of AC and CD in terms of \(s\):
\[ s^2 = AD^2 + \left(\frac{s}{2}\right)^2 \] \[ s^2 = AD^2 + \frac{s^2}{4} \] Rearrange the equation to solve for AD\(^2\):
\[ AD^2 = s^2 - \frac{s^2}{4} = \frac{4s^2 - s^2}{4} = \frac{3s^2}{4} \] We need to express AD\(^2\) in terms of CD\(^2\).
We know that CD = \( \frac{s}{2} \), which implies CD\(^2\) = \( \frac{s^2}{4} \).
Substitute \( \frac{s^2}{4} \) with CD\(^2\) in the expression for AD\(^2\):
\[ AD^2 = 3 \times \left(\frac{s^2}{4}\right) \] \[ AD^2 = 3 \text{CD}^2 \] Step 4: Final Answer:
The value of AD\(^2\) is \( 3 \text{CD}^2 \).